Dynamics of wave packets in the frame of third-order nonlinear Schrödinger equation

Gromov, E. M.; Piskunova, L. V.; Tyutin, V. V.

doi:10.1016/s0375-9601(99)00240-6

Cited by 41 publications

(27 citation statements)

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“…Thus, the above physical argument for semistability does not apply. Numerical results suggest that the families of double-hump embedded solitons in Eq.(1) are still semi-stable [9], but the family of single-hump embedded solitons are fully stable [14].In this paper, we show that single-hump embedded solitons in Eq. (1) are fully stable when |β − 6| ≪ 1 and |µ| ≪ 1.…”

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confidence: 68%

“…In the physical case β = µ, a continuous family of single-hump embedded solitons also exists [8]. In addition, Gromov, et al's numerical computations show that a sech pulse tends to one or a few single-hump embedded solitons [14]. This numerical evidence, together with the above analytical results and physical arguments, strongly suggests that embedded solitons in this physical case are also stable.…”

mentioning

confidence: 83%

“…(1) are still semi-stable [9], but the family of single-hump embedded solitons are fully stable [14].…”

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confidence: 99%

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Stable Embedded Solitons

2003

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Stable embedded solitons are discovered in the generalized third-order nonlinear Schrödinger equation. When this equation can be reduced to a perturbed complex modified KdV equation, we developed a soliton perturbation theory which shows that a continuous family of sech-shaped embedded solitons exist and are nonlinearly stable. These analytical results are confirmed by our numerical simulations. These results establish that, contrary to previous beliefs, embedded solitons can be robust despite being in resonance with the linear spectrum. [PACS: 42.65.Tg, 05.45.Yv.] Pulse propagation in optical fibers attracts a lot of attention these days. Pico-second pulses are well described by the nonlinear Schrödinger (NLS) equation which accounts for the second-order dispersion and self-phase modulation. But for femtosecond pulses, other physical effects such as the third-order dispersion and selfsteepening become non-negligible. The physical model which incorporates these additional effects is [1,2] (1)Here ψ is the envelope of the electric field, z is the distance, τ is the retarded time, β and µ are the selfsteepening coefficients [1]. All quantities have been normalized. For optical pulses, β = µ [1,2]. In this paper, we allow β and µ to be different for the sake of mathematical analysis. The Raman effect, which is dissipative in nature, is also non-negligible [1,2]. It is not included in the model (1) because we want to focus our attention on the other physical effects in this paper. The third-order dispersion term in Eq.(1) is significant because it qualitatively changes the linear dispersion relation of Eq. (1). Its effect on the NLS soliton is to generate continuous-wave radiation and causes the soliton to decay [1][2][3][4][5]. However, solitary waves of Eq. (1) which are embedded inside the linear spectrum do exist in certain parameter regimes [6][7][8][9], and such waves are called embedded solitons [10,11]. To see why a solitary wave of Eq. (1) has to be an embedded soliton, we substitute solitary waves of the form Ψ(τ − V z)e iλz into (1), where velocity V and frequency λ are constants. We readily find that for any frequency λ, the linear equation for Ψ allows oscillatory solutions. Thus, all λ lies in the continuous spectrum of Eq. (1), hence the solitary wave must be an embedded soliton. Stability of these embedded solitons is clearly an important issue. Previous analytical studies have shown that if embedded solitons are isolated in a conservative system, they are at most semi-stable, i.e., the perturbed soliton persists or decays depending on whether the initial energy is higher or lower than that of the embedded soliton [10,12,13]. A physical explanation for it is as follows [10]. If the initial state has energy higher than the embedded soliton, it just sheds extra energy through tail radiation and asymptotically approaches this embedded soliton; but if the initial state has lower energy than the embedded soliton, the energy loss (through radiation) drives the solution away from the embedded soliton, re...

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Stable Embedded Solitons

2003

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“…Soliton solutions in the framework of the extended NLSE with TOD and nonlinear dispersion were found in Refs. [22][23][24][25][26][27][28][29]. In Refs.…”

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confidence: 99%

Damped solitons in an extended nonlinear Schrödinger equation with a spatial stimulated Raman scattering and decreasing dispersion

Gromov

Malomed

2014

Optics Communications

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Dynamics of solitons is considered in the framework of an extended nonlinear Schrödinger equation (NLSE), which is derived from a system of the Zakharov's type for the interaction between high-and low-frequency (HF and LF) waves. The resulting NLSE includes a pseudo-stimulatedRaman-scattering (pseudo-SRS) term, i.e., a spatial-domain counterpart of the SRS term, which is a known ingredient of the temporal-domain NLSE in optics. Also included is inhomogeneity of the spatial second-order dispersion (SOD) and linear losses of HF waves. It is shown that wavenumber downshift by the pseudo-SRS may be compensated by upshift provided by SOD whose local strength is an exponentially decaying function of the coordinate. An analytical soliton solution with a permanent shape is found in an approximate form, and is verified by comparison with numerical results. Keywords: Extended Nonlinear Schrödinger Equation, Damped Soliton Solution, StimulatedScattering, Damping Low-Frequency Waves, Linear Loss High-Frequency Waves, Inhomogeneity Highlights >Dynamics of damped solitons is studied in an extended inhomogeneous nonlinear Schrödinger equation. >We consider media with stimulated-scattering damping acting on low-frequency waves and inhomogeneous spatial dispersion. >An approximate analytical solution for damped solitons is found in an explicit form.

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“…Some exact solutions have been constructed analytically [10][11][12][13][14][15][16][17][18]. Particularly, some bright and dark solitons of HNLSE [19][20][21][22][23][24][25][26][27][28][29][30] are reported for arbitrary parameters [31][32][33][34][35], as well as W-shaped solitary wave solutions [27,28]. Among these studies, there are two more general cases, i.e., the Sasa-Satsuma equation [14] and the Hirota equation [15,16], which is investigated in terms of the Painlevé analysis for the HNLSE integrability.…”

Section: Introductionmentioning

confidence: 99%

Grey solitons and soliton interaction of higher nonlinear Schrödinger equation

et al. 2010

Can. J. Phys.

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In this paper, the higher nonlinear Schrödinger equation is solved by the Hirota method. As an example, the exact grey one-and two-soliton solutions in explicit forms are generated analytically under the continuous-wave background. The results reveal that the velocity of the grey soliton is clearly affected by the higher order effects, yet the grey soliton propagates without any change in their shape and intensity. The higher order term and the phase velocity play the important role for the maximum valley of grey soliton, i.e., the intensity of grey soliton. For the black soliton, the velocity of soliton is determined only by the higher order effects. The analysis of the asymptotic behavior of a two grey soliton solution shows the collision is elastic.PACS Nos: 42.81.Dp, 42.65.Tg Résumé : Nous solutionnons ici l'équation non linéaire de Schrödinger à l'aide de la méthode de Hirota. Pour fin d'exemple, nous générons analytiquement sous forme explicite les solutions à un et deux solitons gris sous un fond en onde continue. Les résultats révèlent que la vitesse du soliton gris est clairement affectée par les effets d'ordre élevé, même si le soliton gris se propage sans modification de forme ni d'intensité. Les termes d'ordre élevé et la vitesse de phase jouent un rôle important dans la vallée du maximum du soliton gris, i.e. l'intensité du soliton gris. La vitesse d'un soliton noir est déterminée entièrement par les effets d'ordre élevé. L'analyse du comportement asymptotique montre que la collision entre les deux solitons gris est élastique.[Traduit par la Rédaction]

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Dynamics of wave packets in the frame of third-order nonlinear Schrödinger equation

Cited by 41 publications

References 11 publications

Stable Embedded Solitons

Stable Embedded Solitons

Damped solitons in an extended nonlinear Schrödinger equation with a spatial stimulated Raman scattering and decreasing dispersion

Grey solitons and soliton interaction of higher nonlinear Schrödinger equation

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